Problemsinvolving the conservationofmechanicalenergyarerelatively straightforward.you
mustfrst selectan objectto analyze.You mustthen decideif all the foices on-the object are
conservative.The force of gravity is a conservativeforce; the force of friction and alirost all
otherforcesarenon-conservative.
If all forcesthat do work areconservative,then conservationofmechanicalenergyyields
Ko + PEo: Kt + PEs,whereKo andPEoarethe kinetic andpotentialenergiesat Jne point in time
andKl and,Pfu ittethe kinetic andpotentialenergiesfor anotherpoint in iime. you will be given
enoughinformationto calculatethree ofthe four energiesthat appearin the energyequationand
canusethe conservationof mechanicalenergyto computethe fourth
If therearenonconservativeforcesacting on the object,then the appropriateexpressionis
WNc= Er* Eo or Wnc = (KEr+ P&) - (KEn+ PEo)
In manyproblemsyou areaskedto find somepmameterthat occursin the expressionfor the
kineticor potentialenergies.
For example,you maybe askedfor theheightoian object,the
speedofan object,or its mass.You first find therelevantenergy(kineiicor potentiat),ihen solve
lor theparameler.
Questions and Example problems from Chapter 6
Question1
A net extemalforce actson a particle.This net forceis not zero.Is this sufficient informationto
concludethat (a) the velocity ofthe particle changes,(b) the kinetic energyof the particle
changes,aad (c) the speedofthe particrechanges?Give your reasoningin eachcase.
(il
7\^p1
-+J
q t
(b)lE
Fno, -lrq d'+ o --d p,,ffihlo@ztfi,u.1*
-h,su o{uprl
aay\ 1n\'e1J€-'l-.'. q ze,r,^,& J
o {"o,,r"Lv-J
tc) lmaI
QuestiontIn the hgurebelow, a block slidesfrom A to c alonga frictionlessramp,and then it passes
tluough horizontalregion cD, wherea frictional force actson it. Is the block's kineti^cenergy
increasing,
decreasing,
or constantin (a) regionAB, (b) regionBC, and(c) regioncD? (d) Is the
block'smechanicalenergyincreasing,decreasing,or constalt in thor" ,"eionr-?
E.
.,]
B
Bffiffiffiffi
cq)
D
(b)
Lc)
(J)
Ll-+-\
...1.
c alur.hi
, c,a.r\a)tn,
\
,
Problem I
To pull a 50 kg crateacrossa horizontai floor at a constantvelocity, a worker appliesa force
directed20oabovethe horizontal,A 25.0 N frictional force opposesthe motion of the crate.As
the cratemoves3.0 m, what is the work doneon the crateby (a) the worker's force, (b) the
kinetic frictional force,(c) the gravitationalforce on the crate,and(d) the normal force?(e) What
is the total work doneon the crate?
FN
: r',o" ex=o
N
Fgrq)o'
X
F-.
ll
LI
F "oeao"
I
-- " n
2 - rr x
\l
F.-ao": f*
m?
=w
V = Fs*oO
6,)
y7 = (),a"on)(a"o-;co.X.D"
t\
W = ( asN)(3,6^)uo4rloo=f, ?tJ1
t))
W= D
l-
"A.x.c,ottatI
t
..-- T(-
/r,tl1o'
f = 2.,0-aN
- 1o'
A 0.075kg arrow is fired horizontally. The bowstringexertsan averageforce of 65 N on the
arrow over a distanceof 0.90m. With what soeeddoesthe arrow leavethe bow?
FO= o 'D,-l5Y*
F=
->
("5N I
W= Yarq V.)
S = O'9O ln
\,/o = oaf r
v-+ 2.
- )1t fita.t^t:
\ Jfib /-rot\
*,
a
W = '/a,r.t
Fs
-
t/
vo=faE
Y7= 31-f s
s
o
I
t/d.Vtt
oo"o" =
-
,4
/p, ri Vo
r-*----__..-?
-,
Pt = 7a..Vtt
= | A(l"sNr)(o.qo-)
t-'
(o"ors K2)
ry-u
tn)oa
Problem3
41s5eusfielicopterlifts a 79-kgpersonstraightup by meansofa cable.The personhasan
upwardaccelerationof 0.70m/s' andis lifted from resttlrough a distanceof 11 m. (a) What is
the tensionin the cable?How muchwork is doneby (b) the tensionin the cableand(c) the
person'sweight?(d) Usethe work-energytheoremandfind the final speedofthe person.
.?
\---l
a) l-= ?.3*ro"N
Fs co.O - -> \n/= (s"3rlo>N)(rt-) ws(>-'=?"1*lo'':-
t)
W=
e)
-8.5,615
L6tvo"=
W-- (m9) s r-6180"= ('l1h)(1,zor1t')(rr.n)
J) Wru = KE+-KE.'=Kmvi- a^,iij -----Wrao= '/^nrV+'
v'=q=@-'
Probremn
V7=3.\nfs
putsa shot(weight:71.1 N) that leaveshis handata distanceof 1.52m abovethe
A shot-putter
ground.(a) Find the work doneby the gravitationalforcewhen the shothasrisento a height of
2.13 m abovethe ground.(b) Determinethe changein the gravitationalenergyof the shot.
u;= mg'= ?t"tN
.,=
5 0 /, 51m
o.) W-,trs
t,d.O
\M = (*g) ( t'r- h.),c,oal8o"
h9= l.)3nr
= (tr.lN)
W=
t)
(1")3n - 1,5r-) c'o'rl8o"
-LiV.,l j
LPE = PEo-PE, = mg h+- ngh"
lJ|tr = rog( h+-h.) = (rr- rr-r)( a.r:*-/.se^) =
(i ? q -r-
Problem5
A slingshot.fuesa pebblefiom the top of a building at a speedof 14.0m,/s.The building is
31.0m tall. lgnoringair resistance,
find the speedwith whichthepebblestrikesthe groundwhen
the pebbleis fired (a) horizontally, O) vertically shaightup, and (c) vertically straightdown.
Vo= l9.oo/<
^
r
*''*'\^
#^"
/-\'
?
V.=
't
'
4^-41z
a \-/
/a nvr'* rn%h+ = Txnnvi- ftteh"
h J = ?l.o^
I
h5 -- Dcn
ft''9 h"
. + = 7a
_ rnvl+
/r rqV.r
o
.o\_
V'* lq \'
V^=
'6
il(
rc".b : Jfi6
,r'.,ttu
71p,^r\.,ovy\air4 )d,
-4/:rULU
1'4, b, {"c
probtemi
F4^,h
)
A water-skierlets go of the tory r.opguponleavingthe.endof a junip-lamp at a speedof 14.0m/s.
As the drawing indicates,the skier hasa speedof 13.0m,/sat the highestpoiat ofthe j"-p.
Ignoring air resistance,determinethe skier's heightH abovethe top ofthe ramp atthehtghest
13-Om/s
^
.o\4/,",r.
b.ta1,XP
I
/^
ho=
c un
h+=H
r/
\o-
-
lr,
qV :
V .Y = 1 3 . o:.t /' .
lr
f.
ImV"]* tt,gh, = 7x IQV4 + fc)$ b,
\
,a
ll
,1
7>,r\V,- = /If\V4 + fr'? n+
ll
As b+
Vo^ : vl"
+
( ll.o'/')'ll+ =
I
h.
t
---
( P,o,r/1))
| (t"to */s-)
I
l1.oti)/t
zo
,Vrt r\ w\.ruat"ffi'6 4 ^
^
-4poa h=o
=
vot- V*'
2'g
iH--l-rnq
Problem7
The figure below showsa pendulumof lengthL = 3.5m. It is held sothat the pendulummakes
an angle0o= 35oandthenreleasedfrom rest. Find the speedof the pendulumwhen it is at its
lowestpoint.
,-""
I
|
t^
,
^
\
ftl
Yo = Onls
L -LcaO"
V+= (
L (l-L.to")
t,
I
7 x h V',+ < J T+ f O 4 h r = i/a f\ V, * mO Do
^ 4 o\\tAftl ".
""t
f
Lll'@e")
fl c = ?
(
,11*", w'at
|
\o=
/trrrv/ = N) 2ho
('t-caoo";
=
f
J le,to'7r")(2.5.)(l:ca35")
Problem 8
A skier startsfrom rest at the top ofa hill. The skier coastsdown the hill and up a secondhill, as
the drawingillustrates.The crestof the secondhill is cfucular,with a radiusof r:36 m. Negleot
Aiction andair resistance.What must be the heighth of the first hill sothat the skierjust loses
contact with the crest of the second hill.
ofl.urfa"'a)*ol^rnt
$*,uo
)
); $,i^a t\+"r^ J-"s"'--q-^}"^t
\=o
)^4t4
rl
O^ls
h"= h
t^
h9
'ltr'*
= Utr
V .t
-
r
,t*'
!
?-
q n^ =
lD Vo * r-o
'4
N% nu
-
ll
^u7
!a
nn3-T= ^u7.
,/r
z-{,)
' / r r n V . \ + I'O Q hr
'q\'
+
t,
)
.r 1
tn V(a
--4
t-.=
a.
f, ^- = --=-'
ne\C
rDa) =
rn?
o
\'/x
i
Pa'fant
/-\,aJ""r
4=o
n-
Y: =lE
j4't't4 ^?e" t
mqa
= n\7
trt-
Problem9
A roller coaster(375 kg) movesfrom A (5.00 m abovethe ground)to B (20.0 m abovethe
ground).Two nonconservativeforcesarepresent:friction does-2.00 x 10' J ofwork on the cm,
anda chainmechani5md6s5{3.00 x 104J of work to help the car up a long climb. What is the
changein the car's kinetic energy,AKE : KEr- KEo,from A to B?
:l
\n/^. = (KF+, PE+)- (Kt"' 7E')
= ( KF, -Ke") n (?f<-PE,) = bKE * bPE
\v/ \ / -
\ Y .rf, /4f\
/SKEt /\PE
-
V at Aulrt
//
- ).oo*lD'5
t)
3. ooltoqf
------"--
'=--=--
APF rng(t''+-h")
KF = W3"*u" +Wrn".rr-APF
- 5.oom)
( 3't 5 Ys)(r.ao'6-)(aD.onr
=
5,51* lo{f
::=-
= (-1"oo* o's) * (:.co,loqs)(s"slx 1p{5)
|
AKtr=-{.5)"loq=
Problem 10
A pitcherthrows a 0.140kg baseball,and it approachedthe bat at a speedof 40.0 m/s. The bat
doesWo": 70.0 J of work on the ball in hitting it. Igroring air resistance,detetminethe speedof
the ball after the ball leavesthe bat andis 25.0 m abovethe point of impact.
\ = p.laoKg
- (re,r7e;
\d.,. = E+- E, = ( KFr-PF+)
-- ( [qvi.tzh+) - ('/*^v,'-r'n6;)
Vo= 4o.onfs
h"=O-r
v+=?
\,/V.,.= Ta"rcrvf
* .?\+ -l/a.roVot
|,,' 7c = ).S.o,o
Wn".lo.o
r( r,Vo'-mg \+
7* mvf = Wn. *
T
r
Vt - j 4)At * Vo'* lg t,.
I
rf)
l (ro'os)
O.lLlofg
+ ({o. o *,/s)\- }(Q,lo "ls) ( rs,o^n)
{r= '5"1'rls
Problem11
uniforrnly from restto 20.0 m/s in 5.6 s along a level stretchof road.Ignoring
A car accelerates
friction, dete?minethe averagepowerrequiredto acceleratethe car if (a) the weight of the car is
9.0x 103N, and@)theweightof thecaris 1.4x 104N.
O-ls
1-)
Y
.
t
!o.o n/s
"r
L.
5, Gs
urr-.r\-
4u\
/t-
o
P = 7enqvi-
l/
n-'ro
--2
-
?
r,
b. ft\v+
D=
I
f
P: Yrtq,trr)tr.."'t{
---, l? = 3.3t)D1W
p;:-1
=
ffi.1s"
I p
p r o b t e 1m2
j,t"s
.;t
S, trtO,W
I
The cheetahis oneofthe fastestacceieratinganimals,beca[rseit can go from restto 27 m/s
(about60 mi,&) in 4.0 s. If its massis 110 kg, determinethe averagepower developedby the
cheetahduringthe accelerationphaseof its motion. Expressyour answerin (a) watts and(b)
horseoower.
\/
-
-
V o
n
,/r\
L2 .q /:
r
V9 = ).? m/s
- w//t
l/r"a\C- /nnq4*
W =
-fit^1 w"Or-)<Epv"-^
f = f .os
6 = lloK$
T)
r
/2r ID (-v4
4
-\
- vo
7
a,
4\
l-
'/^(ttoFe)(rr,/')a
)
C\
= I . O r l D '! .W
Y
I
4.o s
w
b) t hp= ?ets.?
=
P
= l.orlolW
/Lsr\=l-t3T
\tqs.rw/
)
'l